In response to this question, you might like to analyse the way in which Katrina is first presented to us in this story and how some obvious character flaws are presented thanks to the standing of her father and the wealth that she has:

She was a blooming lass of fresh eighteen; plump as a partridge; ripe and melting and rosy-cheeked as one of her father's peaches, and universally famed, not merely for her beauty, but her vast expectations. She was withal a little of a coquette, as might be perceived even in her dress, which was a mixture of ancient and modern fashions, as most suited to set off her charms. She wore the ornaments of pure yellow gold, which her great-great-grandmother had brought over from Saardam; the tempting stomacher of the olden time, and withal a provokingly short petticoat, to display the prettiest foot and ankle in the country round.

Clearly her parents' decision to let her dress herself as something of a "coquette" can be criticised, and the way that she is famed for her "vast expectations" has clearly led her to become slightly arrogant in her wealth and the future she can expect to enjoy. The choice of what she wears is clearly very calculated so to "set off her charms" and she clearly knows her beauty and is determined to have it displayed to its best advantage, as her desire to "display the prettiest foot and ankle in the country round" shows. Obviously her parents' child rearing philosophy has produced an arrogant flirt who is determined to use her charms and position in life to her own advantage, as Ichabod Crane discovers to his cost.

I would not say that this was a wise decision. Parents who allow their children to behave in such a way encourage the wrong actions that take place afterward. Such actions of inpropriaty often besmirch middle-class females rather than rich and hoity females. Rich children seem to have leeway with any of their idiosyncrasies. Just look at the Kennedys for examples.